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9 guesses to What A Long Strange Trip 

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378 miles
O.K., I’ll bite: 42 miles?
I’m totally wrong with the previous answer . It’s 289.9 miles…I looked it up. Is that fair?
Break it it 32 parts
1/4=8/32 (24/32 remaining)
Half of remaining is 12/32 (12/32 left)
3/4 of remaining is 9/32 (3/32 left)
(3/32)x=21 miles left to go
x=total distance=224 miles
Oops, I forgot the last day he traveled another 1/3 of remaining distance (1/32 of total), so it’s actually (2/32)x=21 miles, so x (total distance) is 336 miles.
I agree with Crabman’s 2nd answer. Google tells me that’s 540.74 km for those in metric nations.
336
336 Miles from Birmingham to Glasgow:
X = total Distance
Day 1: Traveled 1/4 X, 3/4 X to go
Day 2: Traveled 1/2 * 3/4 X = 3/8 X, 3/8 X to go.
Day 3: Traveled 3/4 * 3/8 X = 9/32 X, 3/32 X to go.
Day 4: Traveled 1/3 * 3/32 X = 1/32, 2/32 or 1/16 X to go
1/16 X = 21
X = 16*21 = 336 miles.
Crabman you are the winner!
336 miles. On the first day I traveled one quarter of the distance (84, leaving 252). On day two, I traveled one half of the remaining distance (126, leaving 126). On day three, I traveled three quarters of the remaining distance (94.5, leaving 31.5). Yesterday I traveled one third of the remaining distance (10.5). I now have 21 miles left to travel. To work these numbers out, it is easiest to start at the end and work backwards. Yesterday I traveled 1÷3 of the distance to leave 21 miles, therefore there were 31.5 miles (=21 x 3÷2) at the start of yesterday. On day 3 I traveled 3÷4 of the distance to leave 31.5 miles, therefore there were 126 miles (=31.5 x 4) at the start of day 3. On day 2 I traveled 1÷2 of the distance to leave 126 miles, therefore there were 252 miles (=126 x 2) at the start of day 2. On the first day I traveled 1÷4 of the distance to leave 252 miles, therefore there were 336 miles (=252 x 4÷3) at the start of the first day.